Stochastic equations arising from test particle problems

Dynamical functions depending on the state of one marked test particle of a classical many-body system are considered. The time evolution is described by differential equations whose coefficients are random and in addition depend on the initial state of the test-particle. To remove this dependence a weak-coupling approximation is used. Due to the finite correlation time of the driving stochastic process different equations for the test-particle propagator are obtained, if a one-time description is used. It is shown that this ambiguity is characteristic for weakly coupled systems and vanishes only in the weak-coupling limit. The generator of the resulting Markovian process consists of the differentiations with respect to the velocity- and position variables up to second order.